Differential Equations for a Space Curve According to the Unit Darboux Vector

Yıl 2018, Cilt: 9, 91 - 97, 28.12.2018

Süleyman Şenyurt , Osman Çakır

Öz

In this work, the differential equation of a differentiable curve is expressed, by making use of Laplace and normal Laplace operators, as a linear combination of the unit Darboux vector defined as C = sinφT + cosφB  of that curve. Later, the necessary and sufficient conditions are given for the space curves to be a 1-type Darboux vector.

Anahtar Kelimeler

Darboux vector, Laplacian operator

Kaynakça

• Alfred, G., Modern Differential Geometry of Curves and Surfaces with Mathematica, Boca Raton, Florida, USA: 2nd ed, CRC Press, 1997.
• Arslan, K., Kocayigit, H., Onder, M., Characterizations of space curves with 1-type Darboux instantaneous rotation vector, CommunKorean Math. Soc., 31(2)(2016), 379–388.
• Chen, B.Y., Ishikawa, S., Biharmonic surface in pseudo-Euclidean spaces, Mem Fac Sci Kyushu Univ Ser A, 45(2)(1991), 323–347.
• Fenchel, W., On the differential geometry of closed space curves, Bull Amer Math Soc, 57(1951), 44–54.
• Ferrandez, A., Lucas, P., Merano, MA., Biharmonic Hopf cylinders, Rocky Mountain J of Math, 28(3)(1998), 957–975.
• Ferus, D., Schirrmacher, S., Submanifolds in Euclidean space with simple geodesics, Math Ann., 260(1982), 57–62.
• Kocayigit, H., Hacisalihoglu, H.H., 1-type and biharmonic Frenet curves in Lorentzian 3-space, Iran J Sci Tech Trans A Sci, 33(2)(2009),159-168.
• Kocayigit, H., Hacisalihoglu, H.H., 1-type and biharmonic curves in Euclidean 3-space, Int Electron J Geom, 4(1)(2011), 97–101.
• Struik, D.J., Lectures on Classical Differential Geometry, Dover, GB: 2nd ed, Addison Wesley, 1988.